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x - Faculty of Computer Science - Technische Universität Dresden

x - Faculty of Computer Science - Technische Universität Dresden

Calculating with and

Calculating with and without p,q (14) squares and roots mod n without p,q • extracting roots is difficult: provably so difficult as to factor a) If someone knows 2 significantly different roots of an x mod n, then he can definitely factor n. (i.e. w 1 2 � w2 2 � x, but w1 � �w 2 � n | (w 1 �w 2)) proof: n | w 1 2 -w2 2 � n | (w1+w 2)(w 1-w 2) � gcd(w 1+w 2, n) is p or q p in one factor, q in the other 181

Calculating with and without p,q (15) Continuation squares und roots mod n without p,q b) Sketch of “factoring is difficult � extracting a root is difficult” proof of “factoring is easy � extracting a root is easy” So assumption : � W � PPA: algorithm extracting a root to show : � F � PPA: factoring algorithm structure program F subprogram W [black box] begin ... call W ... polynomially often call W ... end. 182

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